Subgroups — C₃²×#6204 #14518870

Order 1944.

1076 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×Q₈	C₃	1	1	yes	maximal, characteristic
2	C₃×#6204	C₃	1	36	yes	maximal, direct product factor
3	C₃²×Q₈⋊C₉	C₃	1	3	yes	maximal, complemented
4	C₂×C₃²×3₋¹⁺²	—	4	1	no	maximal
5	C₃⁴×C₄	—	3	1	no	—
6	C₃²×3₋¹⁺²	—	4	1	no	Sylow
7	C₃³×Q₈	—	3	9	no	—
8	C₃³×Q₈	C₃²	1	12	yes	—
9	C₃³×Q₈	C₃²	1	1	yes	characteristic
10	#6204	C₃²	1	81	yes	direct product factor
11	C₃×Q₈⋊C₉	C₃²	1	36	yes	complemented
12	C₂×C₃⁴	A₄	1	1	yes	characteristic
13	C₂×C₃×3₋¹⁺²	—	4	36	no	—
14	C₂×C₃²×C₉	—	4	3	no	—
15	C₃³×C₄	—	3	27	no	—
16	C₃³×C₄	—	3	12	no	—
17	C₃³×C₄	—	3	1	no	—
18	C₃⁴	SL(2,𝔽₃)	1	1	yes	characteristic
19	C₃×3₋¹⁺²	—	4	36	no	—
20	C₃²×C₉	—	4	3	no	—
21	C₃²×Q₈	—	3	36	no	—
22	C₃²×Q₈	3₋¹⁺²	1	9	yes	complemented
23	C₃²×Q₈	C₃³	1	9	yes	—
24	C₃²×Q₈	C₃³	1	4	yes	—
25	Q₈⋊C₉	C₃³	1	27	yes	complemented
26	C₂×C₃³	—	3	9	no	—
27	C₂×C₃³	C₃×A₄	1	12	yes	—
28	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
29	C₂×3₋¹⁺²	—	4	81	no	—
30	C₂×C₃×C₉	—	4	36	no	—
31	C₃²×C₄	—	3	108	no	—
32	C₃²×C₄	—	3	9	no	—
33	C₃²×C₄	—	3	9	no	—
34	C₃²×C₄	—	3	4	no	—
35	C₃³	—	3	9	no	—
36	C₃³	C₃×SL(2,𝔽₃)	1	12	yes	—
37	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
38	3₋¹⁺²	—	4	81	no	—
39	C₃×C₉	—	4	36	no	—
40	C₃×Q₈	—	3	9	no	—
41	C₃×Q₈	C₃×3₋¹⁺²	1	12	yes	complemented
42	C₃×Q₈	C₃⁴	1	1	yes	characteristic
43	C₂×C₃²	—	3	36	no	—
44	C₂×C₃²	#1094	1	9	yes	—
45	C₂×C₃²	C₃²×A₄	1	9	yes	—
46	C₂×C₃²	C₃²×A₄	1	4	yes	—
47	C₂×C₉	—	4	27	no	—
48	C₃×C₄	—	3	27	no	—
49	C₃×C₄	—	3	12	no	—
50	C₃×C₄	—	3	1	no	—
51	C₃²	—	3	36	no	—
52	C₃²	#6204	1	9	yes	direct product factor
53	C₃²	C₃²×SL(2,𝔽₃)	1	9	yes	—
54	C₃²	C₃²×SL(2,𝔽₃)	1	4	yes	—
55	C₉	—	4	27	no	—
56	Q₈	C₃²×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	3	9	no	—
58	C₂×C₃	C₃×#1094	1	12	yes	—
59	C₂×C₃	C₃³×A₄	1	1	yes	characteristic
60	C₄	—	3	1	no	—
61	C₃	—	3	9	no	—
62	C₃	C₃×#6204	1	12	yes	direct product factor
63	C₃	C₃³×SL(2,𝔽₃)	1	1	yes	characteristic
64	C₂	C₃²×#1094	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃³	C₃×A₄
commutator, cosocle	C₃×Q₈	C₃⁴
Frattini	C₂×C₃	C₃³×A₄
Fitting	C₃⁴×Q₈	C₃
prime radical, hypercenter	C₂×C₃⁴	A₄
hypocenter	Q₈	C₃²×3₋¹⁺²
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃³ ◁ C₂×C₃⁴

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×#6204	C₃	direct (N × H)
3. C₃²×Q₈⋊C₉	C₃	semidirect
10. #6204	C₃²	direct (N × H)
11. C₃×Q₈⋊C₉	C₃²	semidirect
22. C₃²×Q₈	3₋¹⁺²	semidirect
25. Q₈⋊C₉	C₃³	semidirect
41. C₃×Q₈	C₃×3₋¹⁺²	semidirect
52. C₃²	#6204	direct (N × H)
56. Q₈	C₃²×3₋¹⁺²	semidirect
62. C₃	C₃×#6204	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃⁴×Q₈	3
C₃×#6204	3
C₃²×Q₈⋊C₉	3
C₂×C₃²×3₋¹⁺²	4

Subgroups — C32×#6204 #14518870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6204 #14518870